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\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-4_234_1003_1007_571} Particles \(P _ { 1 }\) and \(P _ { 2 }\) are each moving with simple harmonic motion along the same straight line. \(P _ { 1 }\) 's motion has centre \(C _ { 1 }\), period \(2 \pi \mathrm {~s}\) and amplitude \(3 \mathrm {~m} ; P _ { 2 }\) 's motion has centre \(C _ { 2 }\), period \(\frac { 4 } { 3 } \pi \mathrm {~s}\) and amplitude 4 m . The points \(C _ { 1 }\) and \(C _ { 2 }\) are 6.5 m apart. The displacements of \(P _ { 1 }\) and \(P _ { 2 }\) from their centres of oscillation at time \(t \mathrm {~s}\) are denoted by \(x _ { 1 } \mathrm {~m}\) and \(x _ { 2 } \mathrm {~m}\) respectively. The diagram shows the positions of the particles at time \(t = 0\), when \(x _ { 1 } = 3\) and \(x _ { 2 } = 4\).

1. State expressions for \(x _ { 1 }\) and \(x _ { 2 }\) in terms of \(t\), which are valid until the particles collide. The particles collide when \(t = 5.99\), correct to 3 significant figures.
2. Find the distance travelled by \(P _ { 2 }\) before the collision takes place.
3. Find the velocities of \(P _ { 1 }\) and \(P _ { 2 }\) immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions.